Construct random matrix with complex eigenvalues that only have negative real part

Question

For an experiment I'm running I want to be able to generate random real $n\times{}n$ matrices that have complex eigenvalues with only negative real part.

Is there any way of constructing these?

Answer 1

Actually this turns out to be very simple.

1. Generate a random $n\times{}n$ matrix $\bf A$.
2. Construct the eigendecomposition $\bf A = \bf Q \bf \Lambda \bf{Q^{-1}}$.
3. Set $\bf \Lambda^-$ identical to $\bf \Lambda$, but with the real part of each element set to have negative sign.
4. Reconstruct $\bf A^- = \bf Q \bf \Lambda^- \bf {Q^{-1}}$.